Magnetic Semiconductors are Frustrated Ferromagnets

Abstract

Starting from microscopic and symmetry considerations, we derive the Hamiltonian describing the exchange interaction between the localized Mn spins and the valence band holes in $Ga_{1-x}Mn_x As$. We find that due to the strong spin-orbit coupling in the valence band, this exchange interaction has a rather complex structure and generates a highly anisotropic effective interaction between the Mn spins. The corresponding ground state has a finite ferromagnetic magnetization but is intrinsically spin-disordered even at zero temperature.